Problem 17 - Chemistry The Central Science In Si Units

Question

(a) How high in meters must a column of ethanol be to exert a pressure equal to that of a \(100-\mathrm{mm}\) column of mercury? The density of ethanol is \(0.79 \mathrm{~g} / \mathrm{mL}\), whereas that of mercury is \(13.6 \mathrm{~g} / \mathrm{mL}\). (b) What pressure, in atmospheres, is exerted on the body of a diver if she is \(10 \mathrm{~m}\) below the surface of the water when the atmospheric pressure is \(100 \mathrm{kPa}\) ? Assume that the density of the water is \(1.00=1.00 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\). The gravitational constant is \(9.81 \mathrm{~m} / \mathrm{s}^{2}\), and \(1 \mathrm{~Pa}=1 \mathrm{~kg} / \mathrm{ms}^{2}\).

Step-by-Step Solution

Answer

(a) The ethanol column must be approximately 1.75 m high. (b) The pressure on the diver is approximately 1.96 atm.

Step 1: Understanding the Pressure Equivalence

Before diving into calculations, we need to understand that the pressure exerted by a column of fluid is determined by the formula \( P = \rho \cdot g \cdot h \), where \( \rho \) is the fluid density, \( g \) is the gravitational acceleration, and \( h \) is the height of the fluid column. We're tasked with finding the height of an ethanol column that exerts the same pressure as a 100-mm column of mercury.

Step 2: Conversion of Mercury Column Height to Meters

Convert the given height of the mercury column from millimeters to meters. Since there are 1000 millimeters in a meter, we have \( 100 \text{ mm} = 0.1 \text{ m} \).

Step 3: Calculating Pressure Exerted by Mercury

Calculate the pressure exerted by the mercury column using \( P = \rho \cdot g \cdot h \) with the density of mercury \( 13.6 \text{ g/mL} = 13600 \text{ kg/m}^3 \). Using \( g = 9.81 \text{ m/s}^2 \) and \( h = 0.1 \text{ m} \), the pressure is \( P_{Hg} = 13600 \times 9.81 \times 0.1 \).

Step 4: Finding the Required Ethanol Column Height

Set the calculated pressure exerted by the mercury column equal to the expression for ethanol to solve for \( h \): \( 1360.56 = 790 \cdot 9.81 \cdot h_{ethanol} \). Solving for \( h_{ethanol} \), we get \( h_{ethanol} = \frac{1360.56}{790 \times 9.81} \approx 1.75 \text{ m} \).

Step 5: Calculating Water Pressure at Depth

For part (b), we need to calculate the pressure exerted by water at a depth of 10 meters using the formula \( P = \rho \cdot g \cdot h \) with the density of water \( 1000 \text{ kg/m}^3 \), \( g = 9.81 \text{ m/s}^2 \), and \( h = 10 \text{ m} \). Thus, \( P_{water} = 1000 \times 9.81 \times 10 = 98100 \text{ Pa} \).

Step 6: Converting Pressure to Atmospheres

The total pressure on the diver is the sum of atmospheric pressure and the water pressure. Convert the atmospheric pressure to pascals: \( 100 \text{ kPa} = 100,000 \text{ Pa} \). Therefore, total pressure in pascals is \( 100,000 + 98100 \). Converting to atmospheres, where \( 1 \text{ atm} = 101325 \text{ Pa} \), we find \( \text{Pressure in atm} = \frac{198100}{101325} \approx 1.96 \text{ atm} \).

Key Concepts

Ethanol Column Height

When considering fluid pressure, the height of a fluid column significantly impacts the pressure exerted at its base. For ethanol, we need to determine how tall the column must be to match the pressure from a mercury column. The key is understanding the formula: \( P = \rho \cdot g \cdot h \).
For ethanol to exert the same pressure as a 100-mm mercury column, we first calculate the pressure from mercury. The given height converts to 0.1 meters (as 1000 mm equals 1 meter). Using the density of mercury (13.6 g/mL), we calculate its pressure. This exact pressure was set equal to the ethanol pressure expression to find the required height for ethanol.
By rearranging the formula, we solve for ethanol's height:

• Set the pressure expressions equal: \[ 13600 \times 9.81 \times 0.1 = 790 \times 9.81 \times h_{ethanol} \]
• Solve for \( h_{ethanol} \) results in approximately 1.75 meters.

This result shows that due to the lower density of ethanol compared to mercury, a significantly taller column is needed to achieve the same pressure level.

Pressure Calculation

Calculating pressure involves knowing how force is distributed over an area. For a fluid column, this is expressed as \( P = \rho \cdot g \cdot h \), where pressure depends on the fluid's density (\( \rho \)), gravitational acceleration (\( g \)), and the fluid column height (\( h \)).
In this context, both part (a) and part (b) of the exercise utilize this basic equation. In part (a), the known quantity is the pressure from a mercury column, simplified to be equal to the pressure from an ethanol column. This puzzle concerns solving for \( h \), the column height for ethanol.
For part (b), diving at a depth means we consider the water pressure. At a depth of 10 meters in water, applying the formula gives us the pressure at that depth. We use the density of water (1000 kg/m³) to find pressure and then add the atmospheric pressure, previously given as 100 kPa.

• Total pressure includes atmospheric and water pressure.
• Conversion to atmospheres reveals that pressures in physiological environments and applications, like diving, are tangible.

Density of Fluids

Density is a crucial concept when dealing with fluid pressure as it directly influences how pressure builds in a fluid column. Density (\( \rho \)) is defined as mass per unit volume, typically given in kg/m³ or g/mL.
In our exercise, we use the densities of mercury and ethanol to understand their roles in pressure calculation. Mercury is noticeably denser than ethanol; \( \rho_{Hg} = 13.6 \) g/mL versus \( \rho_{ethanol} = 0.79 \) g/mL.
This difference explains why a much shorter mercury column can exert the same pressure as a taller ethanol column. Higher density means more mass is concentrated in a given volume, yielding greater pressure when gravity acts on the column.

• Denser fluids exert more pressure for the same column height.
• When comparing different fluids, density is a determinant factor in pressure balancing puzzles.

Knowing and comparing fluid densities allows us to predict how different substances will behave under gravitational forces. This understanding is crucial in fields like hydraulics, engineering, and natural sciences.